RD Chapter 3 Binary Operations Ex 3.3 Solutions
Question - 1 : - Find the identity element in the set I+┬аof all positiveintegers defined by a * b = a + b for all a, b тИИ I+.
Answer - 1 : -
Let e be the identityelement in I+┬аwith respect to * such that
a * e = a = e * a, тИА a тИИ I+
a * e = a and e * a =a, тИА a тИИ I+
a + e = a and e + a =a, тИА a тИИ I+
e = 0, тИА a тИИ I+
Thus, 0 is theidentity element in I+┬аwith respect to *.
Question - 2 : - Find the identity element in the set of all rational numbers except тАУ 1with respect to * defined by a * b = a + b + ab
Answer - 2 : -
Let e be the identityelement in I+┬аwith respect to * such that
a * e = a = e * a, тИА a тИИ Q тАУ {-1}
a * e = a and e * a =a, тИА a тИИ Q тАУ {-1}
a + e + ae = a and e +a + ea = a, тИА a тИИ Q тАУ {-1}
e + ae = 0 and e + ea= 0, тИА a тИИ Q тАУ {-1}
e (1 + a) = 0 and e (1+ a) = 0, тИА a тИИ Q тАУ {-1}
e = 0, тИА a тИИ Q тАУ {-1} [because a not equal to -1]
Thus, 0 is the identityelement in Q тАУ {-1} with respect to *.
Question - 3 : - 
Answer - 3 : -
Question - 4 : - 
Answer - 4 : -